We now consider the convolutionally encoded FTN system, where a uniform interleaver is assumed to be applied between the convolutional encoder and the BPSK modulator.
Suppose a rate-\(R\)  binary terminated convolutional code (CC) \(\mathcal {C}^{\prime }\)  is utilized. We define \(K_b\)  bit streams as the encoder's inputs and \(N_b\)  bit streams as outputs, i.e., \(R=K_b/N_b\) .
According to (REF ), the BER performance can be estimated by considering all possible error sequences, i.e., the distance spectrum.
However, conventional methods, which assume that the all-zero sequence is transmitted, to estimate the BER performance of CC over AWGN channels are not applicable [1]}. This is because the Euclidean distance between coded FTN signals can be different even for error sequences with the same Hamming error weight [2]}.
For the ease of presentation, we refer to the error sequence \(\mathbf {e} = \mathbf {x} - \mathbf {x}^{\prime }\)  after the FTN modulation as the FTN error sequence, and refer to the binary error vector of the difference between two legitimate CC codewords as the CC error sequence.
It is obvious that given the pair of CC codewords, the FTN error sequence and CC error sequence share the same Hammming error weight.
Therefore, we aim to list all the possible CC error sequences and estimate the BER according to the corresponding FTN error sequences.
